Talk:Goodstein sequence
I'm having doubts about whether this is googological. FB100Z • talk • 03:47, September 13, 2012 (UTC) The googological significance is the Goodstein function G(n), which is the number of steps it takes for the Goodstein sequence starting with n to reach zero. The function G(n) grows quite fast, faster than Bowers' extended array function, for instance. Basically, it has the same relavance as the pages for n(3), Tree(3), etc.; it defines large numbers, but doesn't give them interesting names. Deedlit11 (talk) 03:38, September 20, 2012 (UTC) :Point taken. Maybe the article should include some info on the sizes of the numbers generated? FB100Z • talk • 19:49, September 22, 2012 (UTC) :http://mathoverflow.net/questions/93828/how-large-is-tree3 there are talks that G(8) is larger than Dupertri. :Well, I've edited the article; I have a question about LaTeX though. How do you start a new line in LaTeX? I had to go out of math mode each time to get a new line.Deedlit11 (talk) 12:49, September 27, 2012 (UTC) ::Impressive work. I'll do some cleanup when I have the time. FB100Z • talk • 19:37, September 27, 2012 (UTC) I guessed that \(G(n) \approx f_{\varepsilon_0}(n)\) and \(g(n) \approx f_{\omega^\omega}(n)\), and I've added it to the article. I'm also curious about how we could extend the idea of hereditary representations to get Goodstein sequences of length \(f_{\zeta_0}(n)\) or even \(f_{\Gamma_0}(n)\). FB100Z • talk • 22:26, January 26, 2013 (UTC) g(n) has limit ordinal \(\omega^\omega\) in the Hardy hierarchy, but \(\omega\) in fast-growing hierarchy. So \(g(2^n)\) grows as fast as Ackermann function. As for extension to \(f_{\zeta_0}(n)\), there is no problem, we just indicate height of power tower by \(n \rightarrow\ X\), where X is a power tower with n terms high. For example: \(2 \rightarrow 2^2\). Then height of power tower can change: \(3 \rightarrow 3^{3^3}\). We also can have expressions like \(3 \rightarrow 3^{3^3} \rightarrow 3^{3^{3...3^{3^3}}}\), by the way, reducing it into 0 can require about triakulus steps, as I guess. We can go even further if we will indicate length of row of power towers by \(\downarrow\) from above, but expressions become a bit complicated. Ikosarakt1 (talk) 08:32, January 27, 2013 (UTC) Googleaarex, G(5) is precisely \(f_4 (4) - 3\) and G(6) is precisely \(f_6 (6) - 3\), so I believe it is more informative to use Knuth's arrow notation as the bounds. So I've reverted your edit. Deedlit11 (talk) 21:56, December 24, 2013 (UTC) Correction of upper and lower bounds By referring to Caicedo (2007), which is cited in Wikipedia, upper and lower bounds should be corrected to "-2" instead of "-3". For example, G(4)=f_ω(3)-2 = 3·2402653211 − 2. �� Fish fish fish ... �� 08:15, November 14, 2016 (UTC) :Corrected. �� Fish fish fish ... �� 03:51, November 15, 2016 (UTC) ::There appears to be a difference between the definitions given on this article and the Wikipedia article. The definition given on this article defines G(n'') as the number of steps it takes for the Goodstein sequence starting on ''n to reach zero, while the Wikipedia article defines G(n'') as the length of the Goodstein sequence that starts with ''n (which includes the first entry, which is n''). As a result, the outputs of these functions differ by 1. ::Since this discrepancy appears to be caused by a difference in the definition, I reverted your edit for now; however, the edit can be reinstated if for some reason we decide to change the definition of G(''n) on this wiki to that on the Wikipedia article. -- ☁ I want more ⛅ 09:14, November 16, 2016 (UTC) :::OK, now I see the difference. However, the difference is not clear so I will write it in the article. �� Fish fish fish ... �� 10:20, November 16, 2016 (UTC)